A Phenomenological Derivation of the First- and Second-Order Magnetostriction and Morphic Effects for a Nickel Crystal

P H Y S I C A L R E V I E W V O L U M E 8 2 , N U M B E R 5 J U N E 1, 1951 A Phenomenological Derivation of the First- and Second-Order Magnetostriction and Morphic Effects for a Nickel Crystal W. P. MASON Bell Telephone Laboratories, Murray Hill, New Jersey (Received October 23, 1950) In order to account for experimental results which showed that the saturation elastic eonstants of a single nickel crystal varied with the direction of magnetization, a phenomenological investigation has been made of the stress, strain, and magnetic relations for single nickel crystals. The variation in elastic constants is shown to be a "morphic" effect caused by the change in the crystal symmetry due to the magnetostriction effect. In the energy equation this effect is represented by additional terms which involve squares and products of both the magnetic intensities and stresses. These terms are as large as the magnetostrictive terms when the stresses are of the order of 1010 dynes/cm2. The energy equation has been used to derive the first- and second-order magnetostrictive effect, and the resulting terms agree with Becker and Doring's empirical constants for saturation conditions. For smaller magnetic intensities the terms divide up into first- and second-order terms which vary differently with magnetic field intensity. It is shown that the morphic effects involve six measurable constants, and some of these are evaluated experimentally. I. INTRODUCTION CAREFUL measurements of the change in elastic constants with magnetization in single nickel crystals1 have shown that the elastic constants for a saturated crystal depend on the direction of saturation. It is the primary purpose of this paper to show that the change in saturation elastic constants with direction of magnetization is due to what Mueller2 has called a "morphic" effect; namely, that the distortion of the crystal by the magnetostrictive effect produces a crystal of lower symmetry which involves more than the three elastic constants of a cubic crystal. It is shown in this paper that the additional elastic constants and their variation with magnetic orientation can be obtained by adding terms to the energy equation which involve squares and products of both the magnetic intensity and the stresses. Symmetry determines the number of constants, and it is shown that there are six measurable constants which characterize the effect. The complete energy equation has been applied to a derivation of the magnetostrictive effect, and it is shown that first-and second-order terms are involved. At saturation the terms agree with those given previously by Becker and Doring,3 but for lower magnetic in- tensities the terms divide up into first- and second-order terms which vary differently with magnetic field intensity. The experimental results of Masiyama4 for a single nickel crystal are fitted by the phenomenological formula derived here, and a slightly better fit is obtained. The evaluation of some of the first- and second-order terms is accomplished. 1 These measurements were made primarily to determine the "AJ3" effect and the microeddy current effects in single nickel crystals and will be discussed in another paper, "The frequency dependence of elastic constants and losses in nickel," by Bozorth, Mason, and McSkimin. The measurements are reproduced here in order to evaluate the morphic effect. 2 H. Mueller, Phys. Rev. 58, 805 (1940). 3 R. Becker and W. Doring, Ferromagnetismus (Verlag. Julius Springer, Berlin, 1934), page 275. 4 Y. Masiyama, Sci. Rep. Tohoku Imp. Univ. 17, 945 (1928). The tensor method is followed, since this results in a considerable economy of effort and readily allows one to change from one coordinate system to another. The tensors are all of the cartesian type, since only rec- tangular coordinate systems are considered. The method followed is to add terms to the energy equation of a form required to agree with the measured effects and to determine the resulting constants by means of symmetry considerations. H. ENERGY FUNCTIONS FOR MAGNETOSTRICTIVE CRYSTALS It is well known that the changes in dimensions, magnetic fields, and temperature for any body subject to magnetostrictive and magnetic effects can be derived from a thermodynamic function. In order to use the measured magnetostrictive constants, it is better to use the stress, intensity of magnetization, and entropy as the independent variables. To do this we introduce the elastic enthalpy Hi, defined by the equation # i = 1 U 1 ijiOij) (1) where U is the internal energy function and Ty and Stj are respectively the stresses and tensor strains, i.e., \j$(dUi/dXj)-{-dUj/dXi], where u is the displacement. The differential form of Hx is known to be dHx= -SijdTij+HrndIm+ ®d 716 W. P . MASON dition. Use can be made of the fact that nickel has a center of symmetry which results in having all the odd rank tensors equal to zero. Furthermore, nickel can be described as being soft magnetically but hard elasti- cally. Hence, we can neglect all energy terms that involve terms higher than products or squares of the stresses. With these limitations, the elastic enthalpy function Hx can be written in the form 2Hi= — ^ijkiTijTkr^rRijkinoIrJoTijTki "T~ifcx ijmn-L nU n-L ij\-Li ijmnop* mX rU o* p-*- ijj I ***• m n ^ n T ' f t - mnopl nU n± o* p I -ft- mnopqr*- nU rU o* jJ- q± r« v*y These equations hold for a crystal with a large number of domains when the directions of the domains are uncorrelated, for then the components of magnet- ization are independent. For a single domain, the magnetic intensity has a fixed value, the saturation value Jo, and only the direction of magnetism can be changed. If ah a2, az are the direction cosines of the magnetic intensity with respect to the crystal axes, then one has J i=a i / 0 ; /2=a2/o; Iz=otzlo; (5) and there is a relation between the three direction cosines, namely, M A G N E T O S T R I C T I O N A N D M O R P H I C E F F E C T S 717 where 5= ai2a22+«2ia32+of22«32, K^l^l('-2K Tn+6KT12)Io 4+(--3KTlll+15K T 1n)Io %l The first three terms represent the elastic energy, the next six terms represent the energy stored by morphic effects, i.e., by the change of shape of the body, the next five terms represent the energy stored by the first- and second-order magnetostrictive effects, while the last two terms represent the magnetic anisotropy terms measured at constant stress. If we neglect the morphic energy terms and set the five constants for the magneto- strictive terms equal to Bx to J56, the internal energy reduces to the form given by Becker and Doring.4 From the value of the R constants found experimentally, the morphic energy terms are about equal to the magneto- strictive terms for stresses in the order of 1010 dynes/cm2, but are smaller for smaller stresses. The anisotropy energy constants Ki and Ki are those measured for constant stress. As shown by Kittel,6 these include the anisotropy energy for constant lattice separation plus the magnetostrictive energy caused by the lattice dis- tortion. To obtain the tensor strains from this function one differentiates with respect to the corresponding stress, as shown by Eq. (3). For the engineering strain values which are twice as large as the tensor strain values for the shearing components Sn^-ldHx/dTn) » = 4 , 5 ,6 . (9) Hence, the shear elastic constants s744, and other terms which are multiplied by the stresses r 4 , T&, and T* are one-fourth as large as the corresponding terms which express the relationship between engineering strains and stresses. in . MAGNETOSTRICTIVE EFFECTS IN CUBIC CRYSTALS When a nickel crystal is magnetized, it contracts along the direction of magnetization and expands at right angles to the direction of magnetization. The measurements of Masiyama4 are all elongations along crystallographic axes or along directions such as the [110] or [111]. The elongations or contractions for any direction %' can be obtained from the energy expression (8) by differentiation with respect to the longitudinal stress in this direction or Su'=-dHi/dTu'\ (10) but the strain Su is related to the strains referred to the crystallographic axes by the tensor transformation equation dxjdxi' dxz' dxz' dHx ^33 = Sij= , (11) dXi dXj dXi dXj dTi, • C. Kittel, Revs. Modem Phys. 21, 541 (1949). where the partial derivatives are the direction cosines 0i, 02, and 03 between the zr axis and the xy y, and z axes, respectively. In terms of the one index stress and strain terms 5/= - (0!2d#i/c^i) - WdH1/dT2) - WdHt/dTz) - (20203aF1/dr4)~ (20103dffi/ar5) -20102d£T1/5r6. (12) Performing this differentiation, we have for the magneto- strictive terms S3 ' = Al[ 718 W. P. M A S O N 100 X -eo - 6 0 - 4 0 \ 10 K> 110 THEORETICAL 4 CONSTANT FORMULA BECKER ANOOORING 4 CONSTANT EO.UATION • MEASURED POINTS 010 TlO K DECREES ROTATION 120 140 160 160 FIG. 1. Difference between longitudinal and transverse magnetostrictive constants for 001 plane. striction starting from a fully demagnetized crystal, Eq. (13) gives the magnetostriction as a function of the magnetic intensity. Becker and Doring have compared this formula with the measured curves of Masiyama4 for the magneto- striction at saturation for nickel. They compare the difference between the longitudinal and transverse magnetostriction with their formula for the 001 plane, the Oil plane, and the 111 plane. This difference was chosen because it is independent of the domain structure of the demagnetized state. By choosing the values * i = ~ 24X10-*, ^2=-94XlO-6 , A4=-51X10-*, /*6= + 104X10^, hz=0, (17) the dotted curves of Figs. 1, 2, and 3 result for these three planes. Another comparison was made with the experimental data, and it appears that the data are fittec^better with slightly different coefficients. For the 001 plane the equation for Xj—X*, the difference between the longi- tudinal and transverse magnetostrictive constants becomes Xz-A*= (Ai+A4) cos 220+^2 sin 220. (18) The solid curve of Fig. 1 results when *!+*4=-76X10-*, |^2=~46X10-6 . (19) For the Oil plane the equation for X*—X* is h- X, = K6^i+5/*4)[cos 2*- sin2*][cos2*- s i n 2 ^ ] +ih2 sin 2*[7 cos2^+2 sin2!*] + J M i n 2 * cos2*, (20) THEORETICAL 4 CONSTANT FORMULA BECKER AND OORING 4 CONSTANT FORMULA • MEASURED POINTS where * is the angle measured from the 100 direction The best fit with the measured points is obtained by setting K6/h+5/*4)=-70X10- 6, ^5=+54XlO- 8 . (21) The solid line of Fig. 2 results. This still does not fit the measured values completely, which indicates that higher order energy terms are not negligible; but a better agreement is obtained than with the values used by Becker and Doring. If we combine the first value of (19) with the first value of (21), we have the relation Ai=-40X10- A4=-36X10- 6 . (22) For the 111 plane, a calculation of X*— X* shows that the difference is independent of orientation and has the value \i-\=hi/3+2hi/9+h2/3+fo/lS. (23) No new constants are involved, but one check is obtained for the values determined previously. These add up to — 46X10-6, which agrees well with the measured results as shown by Fig. 3. These measurements do not allow one to separate out all of the constants, since we cannot resolve the values of the constants of Eq. (13). The two involving both second- and fourth-order constants, hi and h2, hi=h(Mu-M12)h 2+ti(Nln- -Niu) +§(Nu2-N12z)- •3iV44l]/04 (24) h2= 2MAJo 2+ [42V112+ (xVm- Nnz) ~ 2iV44i]/o 4, could be resolved if the magnetostriction constants were measured in terms of the magnetic intensity, since terms in P and I4 occur. Unfortunately, however, Masiyama's measurements are presented in terms of the field strengths. Two of these measurements for the 001 plane are shown by Fig. 4. In order to obtain the magnetic intensities, one would need to know the permeabilities for the two directions as a function of field strength. The initial slopes of these two curves, however, can be used to determine approximately the ratio of the Mu constant to the Mn—Mu constant. This follows from the fact that the initial permeability is a second-rank tensor and hence has the same value for all directions for a cubic crystal. From the initial slopes it appears that the ratio is or 2Mu/(Mn- Mu) = 69/40 Mu=0M(Mn-M12). (25) FIG. 2. Difference between longitudinal and transverse magnetostrictive constants for Oil plane. Further constants could be evaluated if shear magneto- striction measurements were available. IV. CHANGE OF ELECTRIC CONSTANTS WITH THE DIRECTION OF MAGNETIZATION FOR A SATURATED CRYSTAL When the crystal is magnetically saturated, it is found experimentally that for elastic waves propagated along the [110], [111], and [100] directions the MAGNETOSTRICTION AND MORPHIC EFFECTS 719 velocity depends on the direction of magnetic satura- tion. This can be explained if we include "morphic" effects; i.e., if we take account of the change of shape of the crystal from a cubic form due to the magneto- strictive effects in the crystal. These changes are caused by the R terms of Eq. (8). For a plane progressive ultrasonic wave it is much more advantageous to express the stresses in terms of the strains, since only one strain occurs in an uncoupled plane progressive wave. This involves expressing the results in terms of the internal energy function U rather than the elastic enthalpy function Hi. The resulting terms can be determined by eliminating the stresses fro Hi and replacing them by the strains. The details are discussed in Appendix II . -eox K>W — T. «* 3 < -20 ! 1 L - J C — — — < THEORETICAL 4 CONSTANT FORMULA BECKER AND CORING A CONSTANT FORMULA • MEASUREO POINTS — — — — —M. — S -** —H DEGREES ROTATION FIG. 3. Difference between longitudinal and transverse magnetostrictive constants for 111 plane. If we differentiate ¥ with respect to Sn and neglect the constant forces generated by the magnetostrictive effect, the stress-strain relation can be expressed in the matrix form Ti T4 Tb S i C7 i i+$ii C712+5l2 cru-{-5n Su Su 1 5i6 s2 C712-f-5l2 £7ll-}-$22 C 12+523 524 ^25 5l6 ss C7l2 + 5 i 3 C 12-f~523 c7H+533 524 516 536 S* 5l4 524 524 c744-r-544 545 546 s, 5l5 525 5l5 545 C 7 4 4 + 5 5 5 556 S* 5l6 5l6 536 546 556 C 44~4~5s6 where chu c1^ cTu are the saturated elastic constants and the §'s are the modifications caused by the morphic effects. These are given in terms of the R values in Appendix I I . All the measurements were made on a (110) section and for transmission in this direction the resulting velocities are determined by solving the determinant7 §(cIu+cIu+5u+8m+28u)-pv 2; ^{c1\2+ c I u+8n+ 8 6 6+25i6); itfu+Su+he+he) i(cIn+cIu+h2+du+28u); M ^ n + ^ 4 4 + 5 2 2 + 5 6 6 + 2 5 1 6 ) - p ^ ; M ^ + ^ + ^ e + ^ e ) K5l4+5l5+546+$56) ; J(^24+525+546+^56) ] 1(2^44+$44+5fi5+2$45)-PV> = 0. (26) If we neglect squares of the 5-quantities, the three constant solutions of this equation can be written P^l 2 =K^ll+^12) + ^44+K5ll+522+2512) +566+251 6 (long), pv2^=^(c In-cI12)+l(di1+d22'-28i2) (shear, particle velocity along [ l l o ] ) , PVZ2 = £J44+ i (544+ $55+ 2^45) (shear, particle velocity along [001]). cMBef^=KcIn-cIi2)+i(c I u-'C Ii2)2Io2 Xtia^iXRiu-Rm+lRm-lRinn (29) The measurement allows one to determine the sum of (27) two other combinations. Finally, for the longitudinal velocity vh the elastic FO* HO DIRECTION From the last shear velocity vZl we find that the morphic value of the shear elastic constant is ^ 4 4 = ^ 4 4 + ^ J 4 4 2 / 0 2 [ ( « 3 2 - i ) ( ^ 4 4 1 - i ? 5 5 l ) -2aia2R466J (28) This allows one to derive one relation between two of the independent constants. For the other shear velocity v2y one has an elastic [ M 4 4 j2 / N „ t - N | U \ , 4 l . , . a , , .« FOR 100 DIRECTION [M „ ~ M . , i * N,„ - N„« i « l . / l \ a , / l \ 4 -eo -* $ - 2 0 110 r r 1 1 i I i i 1i // // J 100 _ 110 _ 7 Love, Theory of Elasticity, Fourth Edition (Cambridge Uni- FIG. 4. Longitudinal minus transverse magnetostriction effect versity Press, London, 1928), p. 298. plotted as a function of field strength H for 001 plane. 720 W . P . M A S O N 0 . 024 0.020 0.012 0.008 0 .004 r ) Ji ft fa JI ft It " 7 1 fl fi t 1 °-02?4w ' 2 0 AMPS. H it [no] e H [no] V II [110] V 0 = 2.277 X 1 0 5 C& = 4.615 X I O ^ SAT C 1 = 4 . 8 2 2 AC=2 .07 X10 1 0 3 4 AMPERES FIG. 5. Change in velocity for shear wave No. 1 as a function of magnetization (10 megacycles). constant becomes ^ l l , = | [ ^ 1 1 + ^ 1 2 ] + ^ 4 4 + ( a 3 2 - i ) / 0 2 - J ( c ' n 2 - c V ) (Rm- Rm) - * V ( * 4 « - Rui) \ - 2cIua1QL2Io 2tcIuRu*+ (cIn+cIl2)R2uJ (30) 0.028 r 0.024 0 .020 0.012 0 .008 / f /1 1^ 1 / / \* 1 I) If II rJ- if if i P* V \ H^" H | | | enl 0 . 0 2 6 — • 2 0 AMPS. pot] 3 To] V | | [ H O ] ! V g = 2.277 X105 CM/SEC. C D = 4 .615X10" AC = 2.41XJ0'0 1 2 3 4 I | AMPERES 3 4 0 6 4 0 980 1200 OERSTEDS FIG. 6. Change in velocity for shear wave No. 1 as a function of magnetization (10 megacycles). V. EXPERIMENTAL RESULTS The original measurements8 which indicated the presence of a "morphic" effect were measurements of the velocity and attenuation of ultrasonic waves in single nickel crystals. These were made primarily to determine the "A£" effect in single crystals and will be discussed in detail in a companion paper.1 The values are reproduced here in order to evaluate the magnitude of the morphic effects. All the measurements were made on a (110) section, since it has been shown that three independent waves, two shear and one longi- tudinal, can be propagated in such a section. One shear wave (called No. 1) is_ generated when the particle motion is along the [1103 direction, while the other (No. 2) is generated when the particle motion is along 0 012 0 .008 >< 0 .004 - 0 . 0 0 4 ^ - o - C 7/ 3 11 fi h t \ i i i 6 i i i i i hi If II 1 1 0 . 0 1 3 0 — - 20 AMPS. 1 H ii [no] e II [ooi] v II [no] V 0 = 3.66 X 1 0 5 CM/SEC C = l.t93 X 1 0 , 2 + 3.11 X10 1 0 DYNES/CM2 3 4 AMPERES 8 These measurements were made by H. J. McSkimin by a technique described in J. Acoust. Soc. Am. 24, 413 (1950). FIG. 7. Change in velocity for shear No. 2 as a function of magnetization (10 megacycles). the [001] direction. Figure 5 shows the increase in velocity divided by the velocity for the demagnetized condition as a function of current through the magnet- izing coil for the No. 1 shear when the field is in the direction of particle motion [110]. At saturation the velocity is increased by a factor of 0.0224. The dotted curve shows the velocity under decreasing conditions. The velocity at zero field is less than that for the demagnetized case, but returns to it when the crystal is again demagnetized. Figure 6 shows a measurement for the same shear with the field parallel to the [001] direction. For this case the increase is 0.026 giving an increase in elastic constant Ac of 2.41 X1010 dynes/cm, compared with 2.07X1010 dynes/cm2 for the field in the [110] direction. The difference between these of 3.4X 109 dynes/cm is a morphic effect. Figures 7 and 8 M A G N E T O S T R I C T I O N A N D M O R P H I C E F F E C T S 721 show similar measurements for the No. 2 shear as a function of magnetic orientation. Here the morphic effect is 2.5 X109 dynes/cm2. Figures 9 and 10 show similar measurements for the longitudinal wave, giving a morphic effect of 0.9X109 dynes/cm2. All these effects are gathered together in Table I, which shows also the combinations of constants involved. The measurements give three relations between the six measurable combinations. While these are not enough to evaluate all of the constants, they do show the existence of a morphic effect. It is interesting to observe that the changes measured are in the same order of magnitude as the change in the elastic constant occasioned by a temperature expansion which produces a distortion comparable to the magnetostrictive distor- 0.012 O.OJO 0.006 > * 0 .004 0 .002 - 0 . 0 0 2 - 0 . 0 0 4 / / K / | / 4 i t k. «r < / / 1 h 11 \ 1 / / f / • X i 1 l 1 I 1 ) f , II- - i • — 1 ' 0.0140 „ 2 0 AMPS. H l l [00l ] e ii [oot] V II [110] V 0 = 3 . 6 6 X I05 CM/SEC C = 1.!93 X10 '2 + 3 .36X1010 0YNES/CM2 1 2 3 4 5 6 7 AMPERES FIG. 8. Change in velocity for shear wave No. 2 as a function of magnetization (10 megacycles). tion. Since the temperature expansion coefficient of nickel is 12 parts in 106 per degree C, an increase in temperature of 4°C will produce an expansion as large as the magnetostrictive effect. This increase in temper- ature will cause a decrease of the elastic modulus of 0.14 percent9 or a change of 3X109 dynes/cm2 in Young's modulus, which is intermediate between the values measured for the longitudinal and shear effects. While no direct comparison can be made between magnetostriction effects and temperature effects, since one causes a change in volume and the other does not, the fact that they produce effects of the same order of magnitude is indicative of the related nature of the effects, i.e., a separation of adjacent molecules. 0.007 0.005 0.004 0.003 0 002 0.001 -0.001 -0.002 V 1 k Ii J I 1 ' I so L o - i , I* / ' / 1' / \l ,« . ^ ' -- i 1 1 i — - i 0.0061 2 0 A M P S — * - 1 1 H II [110] v ii [no] V o = 6 . 0 1 X10S C° = 3.215X10 722 W . P . M A S O N TABLE I. Measured velocities and elastic constants for a 110 section of a nickel crystal as a function of magnetic orientation. Crystal No. 1 mode Velocity demagnetized Constants determined Long, particle velocity 6.01 X105 along 110 cm/sec Shear particle velocity along 001 Shear particle velocity along llO 3.66X106 cm/sec 2.277X106 cm/sec J along 001; Ac=4.06Xl010 dynes/cm2 / along llO; Ac=3.97Xl010 dynes/cm2 / along 001; Ac = 3.36Xl010 dynes/cm2 / along llO; Ac=3.1lX1010 dynes/cm2 / along 001; Ac=2.4lXl010 dynes/cm2 / along llO; Ac = 2.07Xl010 dynes/cm2 c^11(001)-c J , fn(110)=9X108 dynes/cm2 = /o2CCi(c7ii2-c /i22)~^12 2](i?ui~i?ii2) +cIu(cIi2Riu+(cIu+cIn)R2u)~] cMu(001)-cMu(ilQ) = 2.5X 109 dynes/cm2 — CJ442L#441"~ #551 — i?456l^02 cM,6(001) --c*M(l l0) = 3.4X 10 9 dynes/cm2 = Kc /u-c /i2) 2[i?ni-i?ii2+2(i2123-i?i2i)]/o 2 interchanged in Nanopn and w, 0, p> and z can be interchanged without affecting the values of the terms, there are only 90 inde- pendent terms for even the most unsymmetrical crystal, a triclinic crystal. For the R tensor, i and j , k and I, nt and 0, can be inter- changed leaving only 216 independent terms. Furthermore, from the definition of Rijkino, it is obvious that * and j can be inter- changed with k and I. This reduces the number of independent constants to 126 for Rijkimno- Since the number of independent constants of a sixth-rank tensor10 does not seem to have been worked out for a cubic crystal of class Oh, it is the purpose of this appendix to derive the con- stants. This can be done by applying the symmetry conditions for this crystal in conjunction with the transformation equations for a sixth-rank tensor dXi' dx/ dXk' dXl' dXn' dX0' Ki'i'k'i'n'o' = — r— -r— — —Kijkino, KSl) dXi dXj dxjt dxi dxn dx0 where dxi/dx%, • • •, dx0'/dx0f the partial derivatives, are the direction cosines h to n%. For a cubic crystal of class Td=43*w, the symmetry conditions are x*=—x; y= — y; z——z(iytn) and x=*y\ y~z\ z—x(S); (32) that is, if the x axis is shifted 180° to the — x axis, the constants remain unchanged, etc. The symmetry Oh is similar to this except that a center of symmetry is added. This does not affect polar properties of even order, and hence for a sixth-rank tensor class Ok is equivalent to class T* given by Eq. (32). The simplest conditions to apply is that a 180° orientation around the x, y, and z axes results in the same elastic constants as existed without orientation. For a rotation around the z axis of 180°, the direction cosines are d * i 7 d s i = / i = - l ; d*j7d*i = /i = 0; dx 37d*i=/ 3=0; dxi/dx2—mi==0; &X2 I&X2—ni2 — — 1 ; 8 ^ 3 7 ^ 2 = w 3 = 0 ; dx2/dxz—ti2—0 dxz/dXi=*ni—l. Applying this transformation, all terms for which 1, 2, or 3 occur an even number of times have the same sign and hence are not changed. However, if the terms 1 or 2 occur an odd number of times, the sign of the term is negative, and hence such terms must be equal to zero. If we apply the 180° transformation around x and y also, terms 1, 2, and 3 occurring an odd number of times disappear. This is equivalent to an orthorhombic crystal of class 222 or Z>2. If we replace the six index symbols by three index symbols, such that 11 = 1, 22 = 2, 33 = 3, 23 = 4, 13 = 5, 12 = 6, the remaining terms are Rm #211 #121 #131 #221 #231 #311 #321 #331 0 0 0 0 0 0 0 0 0 #113 #123 #133 #213 #223 #233 #321 #333 0 0 0 0 0 0 #313 0 0 0 0 0 0 0 #515 0 0 0 0 0 0 0 0 #441 0 0 0 0 0 #443 0 0 0 0 0 0 0 #625 #61 0 0 R* 0 0 0 0 #661 0 0 0 0 0 #553 0 o I 0 0 0 0 #661 0 0 0 0 0 #663 #112 # i : # i : #212 #222 #232 #312 #322 #332 0 0 0 0 0 0 0 0 0 0 #155 #255 #355 0 0 0 0 I! 0 1 0 #465 0 o 1 0 0 0 0 #442 0 0 0 #!< 0 0 0 #244 0 0 0 #844 #414 #424 #434 0 0 0 0 0 0 0 0 0 0 0 0 0 #652 0 0 0 0 0 0 #654 0 0 0 0 0 #662 0 0 0 0 #564 0 0 0 #166 0 0 #266 0 0 i?36fl 0 i?456 0 #546 0 0 #816 #626 #636 0 0 0 (33) Hence, for an orthorhombic crystal there are 60 independent terms if RabcT^Rbac For the tensor #0&c of Eq. (4), since Rabc—Rbae, the number of independent terms is 39. From the second condition of (32) we have in addition the 10 Sixth-rank tensors of the elastic type have been considered by F. Birch, Phys. Rev. 71, 809 (1947); but these are not general enough for the Raui™ tensor, since only the ij and kl terms can be interchanged rather than all three sets. symmetry that a rotation of 90° around x, y or z results in the same elastic constants as in the unrotated crystals. These three rotations are given by the direction cosines: x rotation y rotation z rotation ( 1 0 0 \ / 0 0 - 1 \ / 0 1 0 \ 0 0 1 1 1 0 1 0 1 ( - 1 0 0 1. (34) 0 - 1 0 / VI 0 0 / V 0 0 1 / M A G N E T O S T R I C T I O N A N D M O R P H I C E F F E C T S 723 Applying these symmetries to the remaining constants, there are number of relations between the constants. These are Rill = ^222 = ^838, Rll2 =» RlU = -&221= R-221 = RzZl = Ru2, JRm= Rui—^212= jRi22sss Rm9 Rzii = -^138 = i?313 = Rn2 = i?322 = -^233 = ^323, •&123 = -&132 = -^213 — ̂ 312 — ^231 = J&321, Rui = -̂ 552 — ̂ 663, (35) Rthl = RuZ = i?442 s=s i?553 s ^861 = ^682, i ? i 4 4 = R m = R 2 U = -^625 • J R 8 6 6 = i?83«, -&165 = -^515 = i?165= ^ 6 1 6 = Rui = «&424 = .^288 = i^626:=s: i^344= ^ 4 3 4 = i?365 = i?535, i?455= i?546=s i?465 = ^«45== -^664 = i?864- Hence, there are nine remaining independent constants for the R tensor. These relations are used in Eqs. (8). For the # tensor, since the last four numbers can be interchanged, there are three relations between the 9 constants and the number reduce to six. The relations are (36) (37) #112 = #111122 = #111212 = #121 = #188 = #155 J #123 = #112323 = #144. The remaining constants are then #111 , #112 , #123, #441 , #561 , a n d #456. This is the same number and type as in Birch's sixth-rank tensor,10 but the order of some of the subscripts is different. Finally, for the tensor Kmn0pqr of Eq. (4), since all of the six subscripts can be interchanged separately, there are three more relations between the constants and there are only three inde- pendent constants. These relations are KlU = iT65i J -^128 = iT441 == i^458 J and there are only three independent constants, ^n i i Kin, and Km. (38) (39) APPENDIX II. DERIVATION OF TERMS IN INTERNAL ENERGY FUNCTION The method for transforming the expression for Hi into the potential function Z7, the internal energy is one of solving for the stresses in terms of the strains and magnetic intensities and substituting for these in Hi. For the strains we have Sn=-dHi/dTn or 5i=(^1i+A11)ri+(^i2+A22)r2-f-(5 /i2+A33)ra4-A14r4 + A 1 6 r 6 + A 1 « r 2 - f A i ( a 1 8 - i ) - f ^ [ a i 4 + ^ - J ] + A 3 ( i ~ 5 ) (40) 56=Ai«r1+A1«r2+As«r,-f-A46r4+A56r5+(^ ir44+A86)re -f- & 2 a i « 2 + &5«ia2a3*, where A n = ( a i 2 - i ) / o 2 ( J ? l l l - i ? 1 1 2 ) ; • ' • ; A 6 6 = ( a 3 2 - J ) / 4 ) 2 ( J R 4 4 1 - i ? 6 6 l ) . If we solve these for the stresses, we find that the stresses T\ to Tt can be expressed in terms of the strains Si to S* given by S1'=Si-Zhi(*i*~i)+hi(ai*+&--i)+fa{i--s)l (41) St—St—p?2aia2-r" h$a ia**3 2] according to the relation Si' 1 t:Jn-Mii £712+$12 C713+$13 $u $15 | $16 5S' CJ12+5l2 C7 l l+$22 C712+$23 §24 $26 5l6 &' C J U + * 1 I C712+528 c7n4-833 524 $16 $36 5 / $14 §24 $24 C744+$44 $45 $46 •SV $15 $26 $15 $45 £744+$65 $58 S t ' $16 $16 $38 $46 $66 C744+$66 Ti T2 Tz r4 T6 where any cla is equal to cUi**{~\Y+i*3P/*s, (43) where As is the determinant of Eq. (40) and As*> the minor obtained by suppressing the ith row and jth. column. Since the Aty terms of Eq. (40) are all very small, it is per- missible to neglect terms having products or powers of these small quantities. With that restriction one finds that the $»/ values are $ H = - 2 a 2 a 3 / ^ 7 4 4 C c 7 l 1 i < ! u 4 + 2 < ; 7 1 2 J R 2 4 4 l $15= $36= — 2aeia3/o2^744C(^ /ll+^ /12)i?244 + C712i2l44], $16 = $28= — 2tt ia2/o% 7 44[(6 / l l+C / 12) i£244+£ / 12^144] , $24= $34= ~ 2a2«3/o 2 6 7 44[(c 7 l l+C 7 l2 ) i?244+£ 7 12.Ku4] , $25=-2aia3/oVJ44[c7iii2i44-f-2c;7i2i2244], $36= — 2a 1 a2/o 2 £ / 44[c / l l - f t l44+2c 7 12i?244l $44= - C / 4 4 2 ( a i * - i ) / 0 2 ( i ? 4 4 1 ~ - ^ 5 5 l ) , $65= - C 7 4 4 2 ( a 2 2 - i ) / 0 2 ( i ? 4 4 1 ~ i f 5 5 l ) , $68= - * 7 4 4 2 ( « 3 2 - §W(RU1-Rm), $45 •• — C7442«l«2/o2i?456, $46 = — C* ifaiCtzI #RAM, $66= —CI4£a2